10.1 Probabilities of One Event
143
“essential” means “reported in the literature”, and in some cases “collected expert
opinion” together with an encoding methodology that makes it possible to transform
“knowledge” into a probability.
10.1.1 Initial Estimates
Appendix A shows how the probability of failure of tailings dams and other events
scales with respect to many real-life examples. The tables in Appendix A can be
used to generate first estimates of probability of failure of various events, in relative
terms, when no or very little statistical data and history are available.
By selecting a wide range of probabilities (and consequences) for each event a
risk assessment will become amenable to a Bayesian update of probabilities when
new data become available. Bayesian updates are developed after a first, a priori evaluation is mathematically corrected using the Bayes theorem, as new data become
available. New data become available as monitoring and observations on site progresses and, in some cases, when clients ask for their operations to be monitored by
space observation.
We generally start a QRA (called the a priori assessment) using uniform distributions for probabilities and consequences. The uniform distribution leads to the
most conservative estimate of uncertainty, as it gives the largest standard deviation
(NIST/SEMATECH 2012). The uniform distribution makes it possible to evaluate
first and second moments of functions of stochastic variables “by hand” (by means
of direct formulae or using the point estimate method developed by Rosenblueth
(1975)), making it possible to bypass “black-box” solutions, such as the Monte
Carlo simulation, which, again, give a sense of false precision. Interested readers
can go deeper into the theme of using uniform distributions as a priori distribution
in a Bayesian approach (see Sect. 10.1.2) by reading the literature on uninformative
priors
1 (Carlin and Louis 2008).
On the other hand, in some cases—for example when adding independent random
variables, or considering higher levels of information for a variable (e.g. min-max,
first and second moment, i.e., average and standard deviation) based on the Central
Limit Theorem—it is possible to assume that the result tends toward a normal distribution (informally, a “bell curve”) even if the original variables themselves are not
normally distributed. The Central Limit Theorem implies that probabilistic and statistical methods that work for normal distributions can be applicable (with caution)
to many problems involving other types of distributions.
The next step, as the information level increases, is to use an empiric distribution,
such as a Beta distribution. Finally, when the knowledge increases further, the “real”
distribution of a variable can be determined (exponential, Gumbel, etc.). Each time
1 The term “uninformative” is common but misleading, as the simple knowledge or estimate of
min-max already constitutes a very valuable piece of information.
143
“essential” means “reported in the literature”, and in some cases “collected expert
opinion” together with an encoding methodology that makes it possible to transform
“knowledge” into a probability.
10.1.1 Initial Estimates
Appendix A shows how the probability of failure of tailings dams and other events
scales with respect to many real-life examples. The tables in Appendix A can be
used to generate first estimates of probability of failure of various events, in relative
terms, when no or very little statistical data and history are available.
By selecting a wide range of probabilities (and consequences) for each event a
risk assessment will become amenable to a Bayesian update of probabilities when
new data become available. Bayesian updates are developed after a first, a priori evaluation is mathematically corrected using the Bayes theorem, as new data become
available. New data become available as monitoring and observations on site progresses and, in some cases, when clients ask for their operations to be monitored by
space observation.
We generally start a QRA (called the a priori assessment) using uniform distributions for probabilities and consequences. The uniform distribution leads to the
most conservative estimate of uncertainty, as it gives the largest standard deviation
(NIST/SEMATECH 2012). The uniform distribution makes it possible to evaluate
first and second moments of functions of stochastic variables “by hand” (by means
of direct formulae or using the point estimate method developed by Rosenblueth
(1975)), making it possible to bypass “black-box” solutions, such as the Monte
Carlo simulation, which, again, give a sense of false precision. Interested readers
can go deeper into the theme of using uniform distributions as a priori distribution
in a Bayesian approach (see Sect. 10.1.2) by reading the literature on uninformative
priors
1 (Carlin and Louis 2008).
On the other hand, in some cases—for example when adding independent random
variables, or considering higher levels of information for a variable (e.g. min-max,
first and second moment, i.e., average and standard deviation) based on the Central
Limit Theorem—it is possible to assume that the result tends toward a normal distribution (informally, a “bell curve”) even if the original variables themselves are not
normally distributed. The Central Limit Theorem implies that probabilistic and statistical methods that work for normal distributions can be applicable (with caution)
to many problems involving other types of distributions.
The next step, as the information level increases, is to use an empiric distribution,
such as a Beta distribution. Finally, when the knowledge increases further, the “real”
distribution of a variable can be determined (exponential, Gumbel, etc.). Each time
1 The term “uninformative” is common but misleading, as the simple knowledge or estimate of
min-max already constitutes a very valuable piece of information.